Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-26089x+1610012\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-26089xz^2+1610012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-33810723x+75218163678\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-162, 1342)$ | $2.2054589111861876288935280859$ | $\infty$ |
| $(99, -50)$ | $0$ | $2$ |
Integral points
\( \left(-162, 1342\right) \), \( \left(-162, -1181\right) \), \( \left(99, -50\right) \)
Invariants
| Conductor: | $N$ | = | \( 479370 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $13561971718800$ | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19 \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3301293169}{22800} \) | = | $2^{-4} \cdot 3^{-1} \cdot 5^{-2} \cdot 19^{-1} \cdot 1489^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3534182836848456390348413541$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33022963130839137455679466208$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8908034309086049$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2202309478844784$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2054589111861876288935280859$ |
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| Real period: | $\Omega$ | ≈ | $0.71044074303778089983742530033$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.2673914704096414905104351093 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.267391470 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.710441 \cdot 2.205459 \cdot 16}{2^2} \\ & \approx 6.267391470\end{aligned}$$
Modular invariants
Modular form 479370.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1548288 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13224 = 2^{3} \cdot 3 \cdot 19 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5015 & 0 \\ 0 & 13223 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7019 & 174 \\ 290 & 59 \end{array}\right),\left(\begin{array}{rr} 10876 & 5017 \\ 5423 & 11862 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 13218 & 13219 \end{array}\right),\left(\begin{array}{rr} 3481 & 3480 \\ 7366 & 7135 \end{array}\right),\left(\begin{array}{rr} 13217 & 8 \\ 13216 & 9 \end{array}\right),\left(\begin{array}{rr} 6236 & 5017 \\ 5191 & 11862 \end{array}\right)$.
The torsion field $K:=\Q(E[13224])$ is a degree-$128989731225600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13224\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 47937 = 3 \cdot 19 \cdot 29^{2} \) |
| $3$ | split multiplicative | $4$ | \( 159790 = 2 \cdot 5 \cdot 19 \cdot 29^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 95874 = 2 \cdot 3 \cdot 19 \cdot 29^{2} \) |
| $19$ | split multiplicative | $20$ | \( 25230 = 2 \cdot 3 \cdot 5 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 479370.z
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 570.g3, its twist by $29$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.